Scientific Notation
Then
Now
Why?
You found products
and quotients of
monomials.
1
2
Space tourism is a multibillion dollar
industry. For a price of $20 million, a
civilian can travel on a rocket or shuttle
and visit the International Space Station
(ISS) for a week.
(Lessons 7-1 and 7-2)
NewVocabulary
scientific notation
Express numbers in
scientific notation.
Find products and
quotients of numbers
expressed in
scientific notation.
1 Scientific Notation
Very large and very small numbers such as $20 million
can be cumbersome to use in calculations. For this reason, numbers are often
expressed in scientific notation. A number written in scientific notation is of the
form a × 10 n, where 1 ≤ a < 10 and n is an integer.
KeyConcept Standard Form to Scientific Notation
i S
OL
Virginia
SOL
A.2.a The student will
perform operations on
polynomials, including
applying the laws of
exponents to perform
operations on expressions.
Step 1 Move the decimal point until it is to the right of the first
nonzero digit. The result is a real number a.
Step 2 Note the number of places n and the direction that you moved
the decimal point.
Step 3 If the decimal point is moved left, write the number as a × 10 n.
If the decimal point is moved right, write the number as a × 10 -n.
Step 4 Remove the unnecessary zeros.
Example 1 Standard Form to Scientific Notation
Express each number in scientific notation.
a. 201,000,000
Step 1 201,000,000
2.01000000
a = 2.01000000
Step 2 The decimal point moved 8 places to the left, so n = 8.
Step 3 201,000,000 = 2.01000000 × 10 8
Step 4 2.01 × 10 8
b. 0.000051
Step 1 0.000051
00005.1
a = 00005.1
Step 2 The decimal point moved 5 places to the right, so n = 5.
Step 3 0.000051 = 00005.1 × 10 -5
Step 4 5.1 × 10 -5
GuidedPractice
1A. 68,700,000,000
416 | Lesson 7-3
1B. 0.0000725
You can also rewrite numbers in scientific notation in standard form.
WatchOut!
Negative Signs Be careful
about the placement of
negative signs. A negative
sign in the exponent means
that the number is between
0 and 1. A negative sign
before the number means
that it is less than 0.
KeyConcept Scientific Notation to Standard Form
Step 1 In a × 10 n, note whether n > 0 or n < 0.
Step 2 If n > 0, move the decimal point n places right.
If n < 0, move the decimal point -n places left.
Step 3 Insert zeros, decimal point, and commas as needed for place value.
Example 2 Scientific Notation to Standard Form
Express each number in standard form.
a. 6.32 × 10 9
Step 1 The exponent is 9, so n = 9.
Step 2 Since n > 0, move the decimal point 9 places to the right.
6.32 × 10 9
6320000000
Step 3 6.32 × 10 9 = 6,320,000,000
Rewrite; insert commas.
b. 4 × 10 -7
Step 1 The exponent is -7, so n = -7.
Step 2 Since n < 0, move the decimal point 7 places to the left.
4 × 10 -7
0000004
Step 3 4 × 10 -7 = 0.0000004
Rewrite; insert a 0 before the decimal point.
GuidedPractice
2A. 3.201 × 10 6
2B. 9.03 × 10 -5
2 Product and Quotients in Scientific Notation
You can use scientific
notation to simplify multiplying and dividing very large and very small numbers.
Problem-SolvingTip
Example 3 Multiply with Scientific Notation
Estimate Reasonable
Answers Estimating an
answer before computing
the solution can help you
determine if your answer
is reasonable.
Evaluate (3.5 × 10 -3)(7 × 10 5). Express the result in both scientific notation and
standard form.
(3.5 × 10 -3)(7 × 10 5)
= (3.5 × 7)(10 -3 × 10 5)
2
= 24.5 × 10
= (2.45 × 10 1) × 10 2
= 2.45 × 10 3 or 2450
Original expression
Commutative and Associative Properties
Product of Powers
24.5 = 2.45 × 10
Product of Powers
GuidedPractice
Evaluate each product. Express the results in both scientific notation and
standard form.
3A. (6.5 × 10 12)(8.7 × 10 -15)
2
3B. (7.8 × 10 -4)
connectED.mcgraw-hill.com
417
Example 4 Divide with Scientific Notation
Evaluate
StudyTip
Quotient of Powers
Recall that the Quotient of
Powers Property is only valid
for powers that have the
same base. Since 10 8 and
10 3 have the same base, the
property applies.
3.066 × 10
_
. Express the result in both scientific notation and
8
7.3 × 10 3
standard form.
( 7.3 ) ( 10 )
3.066 × 10 8
3.066 _
10 8
_
= _
7.3 × 10 3
3
Product rule for fractions
= 0.42 × 10 5
Quotient of Powers
= 4.2 × 10 -1 × 10 5
0.42 = 4.2 × 10 -1
= 4.2 × 10 4
Product of Powers
= 42,000
Standard form
GuidedPractice
Evaluate each quotient. Express the results in both scientific notation
and standard form.
2.3958 × 10 3
4A. __
8
1.305 × 10 3
4B. _
-4
1.98 × 10
1.45 × 10
Real-World Example 5 Use Scientific Notation
MUSIC In the United States, a CD reaches gold status once 500 thousand copies
are sold. A CD reaches platinum status once 1 million or more copies are sold.
a. Express the number of copies of CDs that need to be sold to reach each status
in standard notation.
gold status: 500 thousand = 500,000; platinum status: 1 million = 1,000,000
Real-WorldLink
The platinum award was
created in 1976. In 2004,
the criteria for the award was
extended to digital sales. The
top-selling artist of all time is
the Beatles with 170 million
units sold.
Source: Recording Industry
Association of America
b. Write each number in scientific notation.
gold status: 500,000 = 5 × 10 5; platinum status: 1,000,000 = 1 × 10 6
c. How many copies of a CD have sold if it has gone platinum 13 times? Write
your answer in scientific notation and standard form.
A CD reaches platinum status once it sells 1 million records. Since the CD has
gone platinum 13 times, we need to multiply by 13.
(13)(1 × 10 6)
Original expression
= (13 × 1)(10
6)
Associative Property
= 13 × 10 6
13 × 1 = 13
= (1.3 × 10 1) × 10 6
13 = 1.3 × 10
= 1.3 × 10
7
= 13,000,000
Product of Powers
Standard form
GuidedPractice
5. SATELLITE RADIO Suppose a satellite radio company earned $125.4 million
in one year.
A. Write this number in standard form.
B. Write this number in scientific notation.
C. If the following year the company earned 2.5 times the amount
earned the previous year, determine the amount earned. Write
your answer in scientific notation and standard form.
418 | Lesson 7-3 | Scientific Notation
Check Your Understanding
Example 1
= Step-by-Step Solutions begin on page R12.
Express each number in scientific notation.
1. 185,000,000
2. 1,902,500,000
3. 0.000564
4. 0.00000804
MONEY Express each number in scientific notation.
5. Teens spend $13 billion annually on clothing.
6. Teens have an influence on their families’ spending habit. They control about
$1.5 billion of discretionary income.
Example 2
Express each number in standard form.
7. 1.98 × 10 7
8. 4.052 × 10 6
9. 3.405 × 10 -8
Example 3
10. 6.8 × 10 -5
Evaluate each product. Express the results in both scientific notation and
standard form.
11. (1.2 × 10 3)(1.45 × 10 12)
13. (5.18 × 10 2)(9.1 × 10 -5)
Example 4
12. (7.08 × 10 14)(5 × 10 -9)
14. (2.18 × 10 -2)2
Evaluate each quotient. Express the results in both scientific notation and
standard form.
1.035 × 10 8
15. _
4
2.542 × 10 5
16. _
-10
1.445 × 10
17. __
5
2.05 × 10 -8
18. _
-2
2.3 × 10
4.1 × 10
-7
1.7 × 10
Example 5
4 × 10
19. AIR FILTERS Salvador bought an air purifier to help him deal with his allergies.
The filter in the purifier will stop particles as small as one hundredth of a micron.
A micron is one millionth of a millimeter.
a. Write one hundredth and one micron in standard form.
b. Write one hundredth and one micron in scientific notation.
c. What is the smallest size particle in meters that the filter will stop? Write the
result in both standard form and scientific notation.
Practice and Problem Solving
Example 1
Extra Practice begins on page 815.
Express each number in scientific notation.
20. 1,220,000
21 58,600,000
22. 1,405,000,000,000
23. 0.0000013
24. 0.000056
25. 0.000000000709
EMAIL Express each number in scientific notation.
26. Approximately 100 million emails sent to the President are put into the
National Archives.
27. By 2015, the email security market will generate $6.5 billion.
Example 2
Express each number in standard form.
28. 1 × 10 12
29. 9.4 × 10 7
30. 8.1 × 10 -3
31. 5 × 10 -4
32. 8.73 × 10 11
33. 6.22 × 10 -6
connectED.mcgraw-hill.com
419
Example 2
INTERNET Express each number in standard form.
34. About 2.1 × 10 7 people, aged 12 to 17, use the Internet.
35. Approximately 1.1 × 10 7 teens go online daily.
Examples 3–4 Evaluate each product or quotient. Express the results in both scientific notation
and standard form.
36. (3.807 × 10 3)(5 × 10 2)
9.6 × 10 3
37. _
-4
2.88 × 10 3
38. _
-5
39 (6.5 × 10 7)(7.2 × 10 -2)
40. (9.5 × 10 -18)(9 × 10 9)
8.8 × 10 3
41. _
-4
9.15 × 10 -3
42. _
6.1 × 10
43. (2.01 × 10 -4)(8.9 × 10 -3)
44. (2.58 × 10 2)(3.6 × 10 6)
5.6498 × 10 10
45. __
4
1.363 × 10 16
46. _
6
47. (9.04 × 10 6)(5.2 × 10 -4)
2
48. (2.3 × 10 -3)
6.25 × 10 -4
49. _
2
3.75 × 10 -9
50. _
-4
2
51. (7.2 × 10 7)
8.6 × 10 4
52. _
-6
2
53. (6.3 × 10 -5)
1.2 × 10
2.9 × 10
4 × 10
8.2 × 10
1.25 × 10
1.5 × 10
2 × 10
Example 5
1.2 × 10
54. ASTRONOMY The distance between Earth and the Sun varies throughout the year.
Earth is closest to the Sun in January when the distance is 91.4 million miles. In
July, the distance is greatest at 94.4 million miles.
a. Write 91.4 million in both standard form and in scientific notation.
b. Write 94.4 million in both standard form and in scientific notation.
c. What is the percent increase in distance from January to July? Round to
the nearest tenth of a percent.
B
Evaluate each product or quotient. Express the results in both scientific notation
and standard form.
55. (4.65 × 10 -2)(5 × 10 6)
2.548 × 10 5
56. _
-2
2.135 × 10 5
57. _
3.5 × 10 12
2
58. (3.16 × 10 -2)
5.184 × 10 -5
60. __
3
2
59. (1.4 × 10 6)
61. (5 × 10
3)(
1.8 × 10
2.8 × 10
-7)
7.2 × 10
1.032
× 10 -4
62. __
8.6 × 10 -5
LIGHT The speed of light is approximately 3 × 10 8 meters per second.
63. Write an expression to represent the speed of light in kilometers per second.
64. Write an expression to represent the speed of light in kilometers per hour.
65. Make a table to show how many kilometers light travels in a day, a week, a
30-day month, and a 365-day year. Express your results in scientific notation.
66. The distance from Earth to the Moon is approximately 3.844 × 10 5 kilometers.
How long would it take light to travel from Earth to the Moon?
420 | Lesson 7-3 | Scientific Notation
67 EARTH The population of Earth is about 6.623 × 10 9. The land surface of Earth
is 1.483 × 10 8 square kilometers. What is the population density for the land
surface area of Earth?
68. RIVERS A drainage basin separated from adjacent basins by a ridge, hill,
or mountain is known as a watershed. The watershed of the Amazon
River is 2,300,000 square miles. The watershed of the Mississippi River
is 1,200,000 square miles.
a. Write each of these numbers in scientific notation.
b. How many times as large is the Amazon River watershed as the Mississippi
River watershed?
69. AGRICULTURE In a recent year, farmers planted approximately 92.9 million acres
of corn. They also planted 64.1 million acres of soybeans and 11.1 million acres
of cotton.
a. Write each of these numbers in scientific notation and in standard form.
b. How many times as much corn was planted as soybeans? Write your results
in standard form and in scientific notation. Round your answer to four
decimal places.
c. How many times as much corn was planted as cotton? Write your results
in standard form and in scientific notation. Round your answer to four
decimal places.
H.O.T. Problems
C
Use Higher-Order Thinking Skills
70. REASONING Which is greater, 100 10 or 10 100? Explain your reasoning.
71. ERROR ANALYSIS Syreeta and Pete are solving a division problem with scientific
notation. Is either of them correct? Explain your reasoning.
Pete
Syreeta
3.65 × 10 = 0.73 × 10 -17
_
3.65 × 10 -12
_
= 0.73 × 10 -17
5 × 10 5
-12
5 × 10 5
= 7.3 × 10 -16
= 7.3 × 10 -18
72. CHALLENGE Order these numbers from least to greatest without converting them to
standard form.
5.46 × 10 -3, 6.54 × 10 3, 4.56 × 10 -4, -5.64 × 10 4, -4.65 × 10 5
73. REASONING Determine whether the statement is always, sometimes, or never true.
Give examples or a counterexample to verify your reasoning.
When multiplying two numbers written in scientific notation, the resulting number can
have no more than two digits to the left of the decimal point.
74. OPEN ENDED Write two numbers in scientific notation with a product of 1.3 × 10 -3.
Then name two numbers in scientific notation with a quotient of 1.3 × 10 -3.
75. WRITING IN MATH Write the steps that you would use to divide two numbers
written in scientific notation. Then describe how you would write the results
a
in standard form. Demonstrate by finding _
for a = 2 × 10 3 and b = 4 × 10 5.
b
connectED.mcgraw-hill.com
421
Virginia SOL Practice
A.4.f, A.11
76. Which number represents 0.05604 × 10 8
written in standard form?
A 0.0000000005604
B 560,400
C 5,604,000
D 50,604,000
Distance from
School
77. Toni left school and rode her bike home. The
graph below shows the relationship between
her distance from the school and time.
1
0.75
0.5
0.25
0
y
(30, 1)
(40, 1)
(0, 0)
5 10 15 20 25 30 35 40 45 50 x
Time (minutes)
78. SHORT RESPONSE In his first four years of
coaching football, Coach Delgato’s team won
5 games the first year, 10 games the second
year, 8 games the third year, and 7 games the
fourth year. How many games does the team
need to win during the fifth year to have an
average of 8 wins per year?
79. The table shows the relationship between
Calories and grams of fat contained in an
order of fried chicken from various
restaurants.
Calories
305
410
320
500
510
440
Fat (g)
28
34
28
41
42
38
Assuming that the data can best be described
by a linear model, about how many grams of
fat would you expect to be in a 275-Calorie
order of fried chicken?
Which explanation could account for the
section of the graph from x = 30 to x = 40?
F Toni rode her bike down a hill.
G Toni ran all the way home.
H Toni stopped at a friend’s house on her
way home.
J Toni returned to school to get her
mathematics book.
A
B
C
D
22
25
27
28
Spiral Review
Simplify. Assume that no denominator is equal to zero. (Lesson 7-2)
89
80. _
6
65
81. _
3
8
r 8t 12
82. _
2 7
6
r t
( )
( )
5d 3 g 2 2
( )
3a 4b 4 4
83. _
2
2 4 3
4n p
85. _
3
84. _
4
8c
8p
3h
86. CHEMISTRY Lemon juice is 10 2 times as acidic as tomato juice. Tomato juice is
10 3 times as acidic as egg whites. How many times as acidic is lemon juice as
egg whites? (Lesson 7-1)
Write each equation in slope-intercept form. (Lesson 4-2)
87. y - 2 = 3(x - 1)
88. y - 5 = 6(x + 1)
89. y + 2 = -2(x + 5)
1
90. y + 3 = _
(x + 4)
2
91. y - 1 = _
(x + 9)
1
92. y + 3 = -_
(x + 2)
2
3
4
Skills Review
Simplify each expression. If not possible, write simplified. (Lesson 1-4)
93. 3u + 10u
94. 5a - 2 + 6a
95. 6m 2 - 8m
96. 4w 2 + w + 15w 2
97. 13(5 + 4a)
98. (4t - 6)16
422 | Lesson 7-3 | Scientific Notation